Phase–Curvature Program: Beyond the Riemann Line

Abstract We introduce a dynamical reformulation of the Riemann Hypothesis in terms of a phase–curvature field theory. Starting from the cubic algebra C3 defined by j³ = −1, the phase potential Δ_{C3}(s)—originally a static analytic function—is reinterpreted as a dynamic phase field Φ(σ, t) evolving on the critical strip. A variational principle leads to a C3-anisotropic wave equation, □_{C3}Φ + V′(Φ) = 0, in which the zeros of ζ(s) emerge as topological vortices satisfying quantized phase-flux conditions. This dynamical interpretation bridges number theory and field theory, providing a geometric mechanism for the alignment of nontrivial zeros on the critical line. The framework further opens a path toward a hybrid C6 extension, where multi-resonant phase coupling (π + ζ(3)) generalizes the classical zeta structure. Analytic, algebraic, and numerical components of this Advanced Phase Geometry Program collectively aim to reframe the Riemann Hypothesis as a phase–curvature stability law.